Assessment and optimization for metrology instrument

ABSTRACT

Methods and related program product for assessing and optimizing metrology instruments by determining a total measurement uncertainty (TMU) based on precision and accuracy. The TMU is calculated based on a linear regression analysis and removing a reference measuring system uncertainty (U RMS ) from a net residual error. The TMU provides an objective and more accurate representation of whether a measurement system under test has an ability to sense true product variation.

This application is a continuation of U.S. Ser. No. 10/524,286, now U.S.Pat. No. 7,352,478, filed Feb. 10, 2005, which is a National Stagefiling of PCT/US2002/041180, filed Dec. 20, 2002.

TECHNICAL FIELD

The present invention relates generally to metrology instruments.

BACKGROUND ART

Efficient semiconductor manufacturing requires highly precise andaccurate metrology instruments. In particular, a metrology instrument isrequired to achieve small tolerances to achieve better quality productsand fewer rejections in the manufacturing process. For example, the 1999Edition of the International Technology Roadmap for Semiconductors liststhe necessary precision needed for isolated line control in the year2001 to be 1.8 nm. Unfortunately, correctly assessing and optimizing themeasurement potential of a metrology instrument is difficult for anumber of reasons. For example, an evaluator normally has limited accessto the various instruments under consideration. In addition, eachinstrument needs to be evaluated under a wide range of conditions inorder to gain a valid impression of how it will perform in the actualmanufacturing setting. Finally, there are no widely accepted standardsrelative to the required parameters and how the parameters should bemeasured. As a result, an adequate solution for calculating anuncertainty of a metrology instrument in meaningful units of length forcomparison to manufacturing lithography requirements has been elusive.

Current assessment methods are often based on the repeatability andreproducibility (R&R) of an instrument. For a critical dimension (CD)metrology instrument, evaluation is often executed by pullingrepresentative samples of partially constructed product wafers from amanufacturing line. Recipes (programming instructions) are thenimplemented on an instrument under evaluation such that estimates of thestatic repeatability and long term reproducibility can be made. Forexample, to determine static repeatability for a measurement of a givenproduct level, a recipe is implemented to cause the CD metrologyinstrument to navigate to a particular site on the wafer and thenrepeatedly measure the size of a given feature. The measurementrepeatability is determined from the standard deviation of the acquireddata. Long term reproducibility, also called precision, is determined ina similar way to static repeatability except that between eachmeasurement the sample is removed from the instrument for an arbitrarylength of time ranging from seconds to days. Unfortunately, therepeatability and reproducibility of a measurement is meaningless if themeasurement is wrong. Accuracy must also be considered. Theabove-described methods do not evaluate the accuracy of an instrumentapart from ensuring proper magnification by calibration with pitchstandards. The reason, in part, that accuracy is not considered is thataccepted accuracy standards are generally not available because thespeed at which semiconductor technology advances usually makes anystandard obsolete very quickly. The result of these methodologies isthat a measurement system under test may be misleadingly denoted astrustworthy.

One proposed solution for metrology instrument assessment introduces newparameters related to accuracy in addition to precision. See Banke andArchie, “Characteristics of Accuracy for CD Metrology,” Proceedings ofSPIE, Volume 3677, pp. 291-308 (1999). This approach deviates from usingstandard product wafers as samples by, for example, constructing wafersreferred to as focus and exposure matrix (FEM) wafers. In thismethodology, the actual CD value is determined for various fields on theFEM by using a respected reference measurement system (RMS). Followingthis approach, the RMS values and measurements from the instrument undertest are compared by a linear regression method that is valid forsituations where both variables are subject to error. Use of the FEMwafers is advantageous because they provide examples of productvariation that under normal manufacturing line circumstances may occuronly after a considerable time has passed. Important parameters of thismethodology include the regression slope, the average offset, and a“poorness-of-fit” parameter called nonlinearity. Despite the existenceof this suite of parameters for repeatability, reproducibility andaccuracy, however, an evaluator must still determine, somewhatarbitrarily, how to combine these various parameters to assess oroptimize an instrument.

In view of the foregoing, there is a need in the art for improvedmethods of assessing and optimizing metrology instruments.

DISCLOSURE OF THE INVENTION

The invention relates to methods for assessing and optimizing metrologyinstruments by determining a total measurement uncertainty (TMU) basedon precision and accuracy. The TMU is calculated based on a linearregression analysis and removing a reference measuring systemuncertainty (U_(RMS)) from a net residual error. The TMU provides anobjective and more accurate representation of whether a measurementsystem under test has an ability to sense true product variation.

The foregoing and other features of the invention will be apparent fromthe following more particular description of embodiments of theinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

The embodiments of this invention will be described in detail, withreference to the following figures, wherein like designations denotelike elements, and wherein:

FIG. 1 shows a graph of data for a measurement system under test versusa reference measurement system.

FIGS. 2A-B show flow diagrams of assessment method embodiments of theinvention.

FIG. 3 shows multiple cross-sectional views of an artifact formeasurement.

FIG. 4 shows a graph of data for a couple of CD scanning electronmicroscopes (SEM) under test versus an atomic force microscope (AFM)reference measurement system.

FIG. 5 shows AFM images for one feature of an artifact.

FIG. 6 shows a graph of variation in feature height and sidewall angleacross the various features through photolithographic stepper focus anddose.

FIGS. 7A-B show flow diagrams of optimization method embodiments of theinvention.

FIG. 8 shows a graph of total measurement uncertainty and correctedprecision versus an amount of SEM data smoothing from the optimizationprocess shown in FIGS. 7A-B.

BEST MODE(S) FOR CARRYING OUT THE INVENTION

The description includes the following headings for clarity purposesonly: I. Data Analysis, II. Assessment Method, III. Optimization Method,IV. Conclusion. It should be recognized that while particular types ofmeasurement systems will be mentioned throughout the description thatthe teachings of the invention are applicable to any type of measurementsystem.

I. Data Analysis

In order to determine a total measurement uncertainty (hereinafter“TMU”) of a measurement system under test (hereinafter “MSUT”), it isnecessary to compare measurement data sets of a MSUT and a referencemeasurement system (hereinafter “RMS”). A conventional technique forcomparing such data sets is linear regression derived by plotting thedata sets against one another as shown in FIG. 1. The following dataanalysis is derived from the paper “Characteristics of Accuracy for CDMetrology,” Proceedings of SPIE, Volume 3677, pp. 291-308 (1999) byBanke and Archie, which describes a form of linear regression upon whichthe invention draws. As used herein, precision shall be referred as aone sigma (σ) value.

When regressing one variable onto another an assumption is made aboutthe relationship between the two variables. Referring to FIG. 1, it isassumed that a MSUT, e.g., a CD SEM, should behave linearly to the firstorder when compared to a set of reference standards, i.e., those from anRMS, e.g., a CD AFM. Such a model would be represented by a slope, β,and an intercept, α, like the following equation:y _(i) =α+βx _(i)+ε_(i)  (1)where y_(i) and x_(i) represent the i^(th) dependent and independentvariables, respectively, and ε_(i) is the i^(th) deviation, or residual,from the model. In terms of the metrology instrument assessment andoptimization data analysis and methods, as discussed in more detailbelow, the independent variable x refers to the MSUT and the dependentvariable y refers to the RMS.

The ordinary least-squares (hereinafter “OLS”) fit is one type ofgeneral linear regression analysis, in which no error is assumed in theindependent variable (MSUT). However, there are situations, especiallyin the case of semiconductor industry metrology applications, where thisassumption is not valid. There are criteria that give some indication asto when, or under what conditions, it is permissible to use the OLS. Onecriterion is based upon the precision of the independent variable,σ_(x), being small compared to the standard deviation of all the xvalues:

$\begin{matrix}{\frac{\sigma_{{all} \times {values}}}{\sigma_{x}} ⪢ 1} & (2)\end{matrix}$Another criterion for acceptable use of the OLS fit is:

$\begin{matrix}{{{\beta } \times \frac{\sigma_{x}}{\sigma_{y}}} ⪡ 1} & (3)\end{matrix}$

If the estimated slope is approximately unity, it is easy to see thatthe precision in the independent measurement (MSUT) must be muchsmaller, or better, than the precision in the dependent variable (RMS)for the OLS to be valid. Perhaps most important in testing the accuracyof an unknown MSUT is the effect of the uncertainty in the referencestandards on the resultant parameters that are used to assess thisaccuracy. To account for this, a method of linear regression thataddresses errors in the y (RMS) and x (MSUT) variables and estimates theslope and intercept of the resultant best-fit line is necessary tofairly evaluate the accuracy of a measurement system.

The Mandel linear regression, as introduced in 1964 and revised in 1984by John Mandel, provides a methodology of handling the least-squares fitwhen both variables are subject to error. One of the benefits of thismore generalized regression analysis is that it can be used in alldegrees of error in x and y, even the case when errors in x are zero,σ_(x)=0. One parameter affecting the Mandel method is a variable λ(referred to herein as the “ratio variable”), which is defined by:

$\begin{matrix}{\lambda = \frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}} & (4)\end{matrix}$where σ_(y) and σ_(x) are the precisions of the y (RMS) and x (MSUT)measurements, respectively. In the Mandel method, it is important torecognize that these precisions are based on replication only, notaccuracy. According to the invention, the ratio variable λ is re-definedas:

$\begin{matrix}{\lambda = \frac{U_{RMS}^{2}}{U_{MSUT}^{2}}} & (5)\end{matrix}$where U_(RMS) is an RMS “uncertainty” defined as an RMS precision(σ_(RMS)) or an independently determined RMS total measurementuncertainty (TMU_(RMS)), and U_(MSUT) is an MSUT “uncertainty” definedas a corrected precision of the MSUT or a TMU of the MSUT, as will bemore fully described below. The TMU_(RMS) can be determined using themethods as described herein applied to the RMS, i.e., treating the RMSas an MSUT. Unless denoted “TMU_(RMS),” “TMU” shall refer to the TMU forthe MSUT.

The intent of the Mandel method is to start the analysis of the fittingprocedure with some measure of the confidence level for eachmeasurement. A key metric resulting from this regression is the slope ofthe best-fit line:

$\begin{matrix}{\hat{\beta} = \frac{S_{yy} - {\lambda\; S_{xx}} + \sqrt{( {S_{yy} - {\lambda\; S_{xx}}} )^{2} + {4\lambda\; S_{xy}^{2}}}}{2S_{xy}}} & (6)\end{matrix}$where the S_(xx), S_(yy), and S_(xy) are the sum of the squares from theraw data as defined by:

$\begin{matrix}{{{S_{xx} = {\sum\limits_{i = 1}^{N}( {x_{i} - \overset{\_}{x}} )^{2}}},{S_{yy} = {\sum\limits_{i = 1}^{N}( {y_{i} - \overset{\_}{y}} )^{2}}},{and}}{S_{xy} = {\sum\limits_{i = 1}^{N}{( {x_{i} - \overset{\_}{x}} )( {y_{i} - \overset{\_}{y}} )}}}} & (7)\end{matrix}$where N is the number of ordered pairs. In the general linear regressioncase, where OLS is valid, the uncertainty of the independent variable(MSUT) goes to zero and the ratio variable λ→∞. The estimate for theslope as the ratio variable λ approaches infinity is S_(xy)/S_(xx) andwhen all the error is in the x (MSUT) measurement compared to the y(RMS) measurement, the ratio variable λ approaches zero and the estimatefor the slope is S_(yy)/S_(xy). This would be like regressing x onto y,which points out another feature of the Mandel method of regression. Theanalysis is symmetrical with the x and y variables such that it does notmatter whether x is regressed on y, or y is regressed on x.

Another metric resulting from this methodology is the correctedprecision of a metrology instrument, which is defined as follows:Corrected Precision≡{circumflex over (β)}σ_(x)  (8)As defined, a smaller slope {circumflex over (β)} implies a greaterchange in MSUT measurement for a given change in the RMS values. Use ofa corrected precision is useful because a MSUT could exhibit a smaller(better) precision than other tools under test, yet have a larger(worse) slope. A larger slope would imply a less sensitive measurementtool, while on the other hand, a smaller precision would indicate a moreresolute measurement capable of being sensitive to small changes. Theproduct of these two estimates acts as a balance for the raw,uncorrected, precision. Therefore, for an equivalent corrected precisionof two different MSUTs, a system with a smaller estimated slope{circumflex over (β)} can accommodate a larger precision σ_(x) to yieldan equivalent corrected precision. In other words, the slope correctsthe precision to correspond to the RMS calibrated scale.

As a check and balance on the corrected precision, a specification onthe slope is also required. It is desirable to have a measurement systemwith a unity slope (i.e., slope=1) to maintain a constant offset, whichvaries as a function of the RMS values when the slope is not equal toone. This situation makes for a more complicated correction in amanufacturing environment.

Another parameter of the regression analysis is the estimated intercept,{circumflex over (α)}. This parameter is dependent upon the estimatedslope. As a result, the two parameters of the 1^(st)-order regressionanalysis, i.e., {circumflex over (α)} and {circumflex over (β)}, are notstatistically independent of each other. In addition, since theintercept is a value of y at x=0, it is difficult to get an intuitivemeaning of its value. Instead of this parameter of the regression,another parameter called the offset is used and defined here as:Offset≡Δ= y− x   (9)where x and y are the measurement averages of a calibration effort. Thisparameter is independent of the regression analysis. Recognizing thisand considering that for a calibration effort on a MSUT, itsmeasurements will be regressed against the RMS values, the offset is areflection of the closeness of the MSUT compared to the RMS.

Another check is that the data needs to be tested to see if the x versusy relationship can be described as linear. This check is completed byconsidering the residual error. The residual error definition isdifferent for the general linear regression (e.g., OLS) case compared tothe Mandel case. The residual error for OLS, d_(i), at each ordered pairof data is defined as:d _(i) =y _(i) −{circumflex over (α)}−{circumflex over (β)}x _(i)  (10)where {circumflex over (α)} and {circumflex over (β)} are the estimatedintercept and slope, respectively, of the OLS regression. The netresidual error D is the square root of the mean-squared error of theseresiduals and can be expressed as:

$\begin{matrix}{D^{2} = {\frac{\sum\limits_{i = 1}^{N}d_{i}^{2}}{N - 2}.}} & (11)\end{matrix}$However, this definition of the residual is not correct when the Mandelmethod is applied to the situation of comparing the RMS to the MSUT. Thecorrect net residual error D_(M) is given by:

$\begin{matrix}{D_{M} = \sqrt{\frac{( {\lambda^{2} + {\hat{\beta}}^{2}} )}{( {\lambda + {\hat{\beta}}^{2}} )^{2}}D^{2}}} & (12)\end{matrix}$The net residual error D_(M) is comprised of both systematic and randomcomponents of error. The method of data gathering and analysis describedherein includes accessing the random component of error by replication,creating essentially a precision estimate. Given precision estimatesσ_(x) and σ_(y) for the x (MSUT) and y (RMS) variables, respectively, itis possible to make an estimate of the input variance of the data set:

$\begin{matrix}{{{Var}({input})} = {\sigma_{y}^{2} + {{\hat{\beta}}^{2}{\sigma_{x}^{2}.}}}} & (13)\end{matrix}$

The slope is included in the above definition for reasons similar to itsintroduction into the corrected precision parameter. The ratio of thesquare of the Mandel net residual error D_(M) to the input variance is aparameter that distinguishes systematic error from random error in thedata set. This quantity is referred to herein as the “nonlinearity”parameter:Nonlinearity=D _(M) ²/Var(input)  (14)When the nonlinearity can be shown to be statistically significantlygreater than unity, then the regression is revealing that the datacontains significant nonlinear systematic behavior.

The invention determines a metric referred to herein as “totalmeasurement uncertainty” (hereinafter “TMU”) that summarizes, in aformat directly comparable to measurement requirements, how well theMSUT measures even if its measurements are corrected by the regressionslope {circumflex over (β)} and intercept {circumflex over (α)}. The TMUmetric can be derived from the general linear regression metrics, orpreferably from the Mandel metrics. In particular, TMU can be derivedfrom the Mandel net residual error D_(M). The Mandel net residual errorD_(M) contains contributions from the RMS uncertainty (U_(RMS)), theMSUT uncertainty (U_(MSUT)), and any nonlinearity in the relationshipbetween measurements from these instruments. Similarly, the TMU can bederived from the net residual error D for a general linear regression,which contains contributions from the RMS uncertainty, i.e., in thiscase the RMS precision (σ_(RMS)), the MSUT corrected precision, and anynonlinearity in the relationship between measurements from theseinstruments.

Conceptually, the TMU is the net residual error (D_(M) or D) without thecontribution from the RMS uncertainty (U_(RMS)). TMU assigns to the MSUTmeasurement all other contributions. As noted above, the “RMSuncertainty” (U_(RMS)) is defined as the RMS precision or anindependently determined RMS total measurement uncertainty (TMU_(RMS)).That is, in one instance, U_(RMS) may simply be considered the precisionof the RMS (σ_(RMS)), i.e., σ_(RMS) is used as an estimate of the TMUfor the RMS. However, where the RMS has a TMU substantially differentthan its precision, TMU_(RMS) can be input to the ratio variable λ (Eq.5) for determining the Mandel net residual error D_(M) and the TMUdefinition. The TMU_(RMS) may be independently derived for the RMS,i.e., treating the RMS as a MSUT compared to another RMS. Based onabove, TMU for a Mandel linear regression can be defined as:TMU=√{square root over (D _(M) ² −U _(RMS) ²)}  (15)where D_(M) is the Mandel net residual error. Similarly, TMU for ageneral linear regression, e.g., OLS, can be defined as:TMU=√{square root over (D ² −U _(RMS) ²)}  (16)where D is the net residual error.

It should be recognized, relative to the Mandel linear regression, thatwhen the corrected precision of the MSUT is initially used as the MSUTuncertainty (U_(MSUT)) to calculate ratio variable λ, the subsequentlydetermined TMU value for the MSUT from Eq. 15, in some cases, may besubstantially different from the corrected precision for the MSUT (i.e.,U_(MSUT)). In this case, the linear regression may be repeated with thedetermined TMU value substituted for the corrected precision of the MSUTin the definition of the ratio variable λ (Eq. 5). Similarly, when thesubsequently determined TMU for the MSUT is still substantiallydifferent from the MSUT uncertainty used, the linear regression may berepeated with each new estimate of the TMU substituted for the MSUTuncertainty (U_(MSUT)) in the ratio variable λ (Eq. 5) until sufficientconvergence of the MSUT uncertainty (U_(MSUT)) and TMU is achieved todeclare a self-consistent result.

It should also be recognized that, depending upon the skill with whichthis method is executed and the nature of the measurement techniquesused by the two systems, there may be an undesirable contribution fromthe artifact itself. Properly designed applications of this methodshould minimize or eliminate this contribution.

TMU provides a more correct estimate of the MSUT uncertainty than theprecision estimate alone because it addresses the case where there areerrors due both to precision and accuracy. In contrast, the Mandellinear regression method alone addresses situations where both variablesare only subject to the instrument precisions. Accordingly, TMU is amore objective and comprehensive measure of how the MSUT data deviatesfrom the ideal behavior that would generate a straight-line SL plot inFIG. 1, or the inability of the MSUT to measure accurately. It should berecognized, however, that there are differences between TMU and what isgenerally considered as measurement error, i.e., the quadratic sum ofall possible sources of random and systematic error contributions. Inparticular, systematic errors due to magnification calibration errorsand offset errors are not included in the TMU since, in principle, thesecan be reduced to arbitrarily small contributions given sufficientattention to calibration. TMU represents the limit of what can beachieved for the given type of measurement if sufficient attention ispaid to calibration. As a consequence, it represents a measure of theintrinsic measurement worth of the system.

II. Assessment Method

With reference to FIGS. 2-6, a method and program product for assessinga measurement system under test (MSUT) will be described.

Referring to FIG. 2A, a flow diagram of a method for assessing a MSUTaccording to a first embodiment is shown.

In a first step S1, an artifact for use in assessing the MSUT isprepared. With reference to FIG. 3, “artifact” as used herein shallrefer to a plurality of structures 8 provided on a substrate 16. Anartifact is generated to represent variations in a particularsemiconductor process of interest for the particular MSUT. In oneembodiment, an artifact may be process-stressed samples derived fromactual product. FIG. 3 illustrates exemplary structures for a particularprocess including: an under-exposed structure 10, an ideal structure 12(referred to as the “process of record” (POR) structure), and anoverexposed undercut structure 14. Artifact 8 should be constructed toinclude a fair representation of all of the various scenarios that canarise during manufacturing. The types of artifact provided may varydrastically based on, for example, the type of measurement needingassessment, the manufacturing processes that alter the measurement, andmeasurement parameters that alter the measurement such as temperature,probe damage, manufactured product structure or materials, etc.

Returning to FIG. 2A, at step S2, a critical dimension of artifact 8(FIG. 3) is measured using a reference measurement system (RMS) togenerate an RMS data set. The dimension may include, for example, atleast one of line width, depth, height, sidewall angle, top cornerrounding or any other useful dimension. The RMS is any measuring systemthat is trusted within a particular industry or manufacturing process.The measurement step includes characterizing the artifact(s) andproducing documentation detailing structure location and referencevalues. As part of this step, an RMS uncertainty (U_(RMS)) iscalculated. This calculation may include calculation of an RMS precision(σ_(RMS)) according to any now known or later developed methodology,e.g., a standard deviation analysis. Alternatively, this calculation mayinclude calculating a TMU_(RMS) according to the methods disclosedherein. That is, the RMS may be treated as an MSUT and compared toanother RMS.

At step S3, the same dimension is measured using the MSUT to generate anMSUT data set. This step includes conducting a long-term reproducibility(precision) study of the MSUT according to any now known or laterdeveloped methodology. As part of this step, a MSUT precision σ_(MSUT)from the MSUT data set is also calculated according to any now known orlater developed methodology, e.g., a standard deviation analysis.

Referring again to FIG. 1, a plot of data measured by an MSUT in theform of a CD SEM versus an RMS in the form of an AFM is shown. Asdiscussed in the data analysis section above, if a MSUT is a perfectmeasuring tool, the data sets should generate a straight line (SL inFIG. 1) when plotted against one another, i.e., y=x. That is, the lineshould have unity slope and an intercept at 0 as generated by identicaldata points. However, a MSUT is never a perfect measuring tool becauseit and the artifact are subject to the myriad of process variations. Inmost instances, a 0 intercept or unity slope are unlikely and, evenworse, may have peaks or curvature in the data. All of this representsinaccuracy in the MSUT.

Steps S4-S5 (FIG. 2A) represent calculations of a total measurementuncertainty (TMU) of the MSUT according to the above-described dataanalysis. In a first part, step S4, a Mandel linear regression, asdiscussed above, of the MSUT and RMS data sets is conducted. The Mandellinear regression produces the parameters of slope, net residual errorof the MSUT (i.e., the MSUT data set compared to the RMS data set),corrected precision of the MSUT and average offset.

Next, at step S5, TMU is determined according to the formula:TMU=√{square root over (D _(M) ² −U _(RMS) ²)}  (17)where D_(M) is the Mandel net residual error (Eq. 12) and U_(RMS) is theRMS uncertainty, i.e., the RMS precision (σ_(RMS)) or an independentlydetermined TMU_(RMS). In other words, a TMU for the MSUT is determinedby removing the RMS uncertainty (U_(RMS)) from the net residual errorD_(M).

At step S6, a determination is made as to whether the determined TMU issubstantially different from the MSUT uncertainty (U_(MSUT)). In a firstcycle of steps S4-S5, the MSUT uncertainty is the corrected precision.In subsequent cycles, the MSUT uncertainty is a previously determinedTMU value of the MSUT. If step S6 results in a YES, as discussed above,the Mandel linear regression may be repeated with the previouslydetermined TMU value substituted for the MSUT uncertainty (step S7)(U_(MSUT)) in the ratio variable λ (Eq. 5). The Mandel linear regressionanalysis is preferably repeated until a sufficient convergence of theMSUT uncertainty (U_(MSUT)) and TMU is achieved to declare aself-consistent result. What amounts to “sufficient convergence” or“substantially different” can be user defined, e.g., by a percentage.

If the determination at step S6 is NO, then the determined TMU value isconsidered the final TMU for the MSUT, i.e., sufficient convergence hasoccurred. Based on the final TMU, an objective assessment of the MSUT isachieved.

Referring to FIG. 2B, a flow diagram of a method for assessing a MSUTaccording to a second embodiment is shown. This embodiment issubstantially similar to the embodiment of FIG. 2A, except that thelinear regression can be any general linear regression, e.g., an OLS. Inthis case, the TMU is defined according to the formula:TMU=√{square root over (D ² −U _(RMS) ²)}  (18)where D is the net residual error (Eq. 11) and U_(RMS) is the RMSuncertainty, i.e., the RMS precision (σ_(RMS)) or an independentlydetermined TMU_(RMS). The determined TMU in step S5 is the final TMU.

Asessment Example

Referring to FIG. 4, a graph that compares measurements from two CD SEMs(CD SEM A and CD SEM B) to a respected RMS is shown. The artifact usedwas a focus and exposure matrix (FEM) wafer with the maximum dimensionof an isolated line of resist as the feature of interest. This is aparticularly important geometry and material because it is similar to akey semiconductor processing step that determines the speed with whichtransistors can switch. Hence, tighter and more accurate control at thisstep of manufacturing can produce more computer chips that are extremelyfast and profitable. The RMS in this case was an atomic force microscope(AFM), which was trusted to determine the true CD, namely the maximumlinewidth of the resist.

Ideally this data should lie along a straight line with unity slope andzero offset. The nonlinearity (Eq. 14) parameter characterizes thescatter of the data around the best-fit line. This variance of scatteris normalized so that if all of this variance is due to the randommeasurement variance measured by reproducibility, then the nonlinearityequals unity. In this case, CDSEM A has a nonlinearity of 100 whileCDSEM B has a value of 137. Both are disturbingly large numbers. Thefollowing table derived from this data further illustrates the improvedobjectivity of the TMU parameter.

Total Corrected Measurement Precision Uncertainty [nm] [nm] CDSEM A 1.520.3 CDSEM B 1.8 26.1

This example illustrates the severe discrepancy between using precisionas the key roadmap parameter and TMU, which contains precision but alsoincludes contributions from accuracy. In the particular example of theresist-isolated line, the problem is associated with severe resist lossduring the printing process, which can have profound changes to the lineshape, and how well the MSUT measures the desired critical dimension.FIG. 5 shows multiple AFM images for one of the features on this FEMwafer. The AFM image shows edge roughness, top corner rounding, and evenundercut. Referring to FIG. 6, a graph shows the variation in featureheight and sidewall angle across the FEM. On the horizontal axis is thephotolithographic stepper focus setting. Across this FEM, the featureheight changes by a factor of three (3). In addition, there issignificant sidewall angle variation.

III. Optimization Method

An application for the above-described assessment methodology and TMUcalculation lies in the optimizing of a measurement system. Conventionalmethods for optimizing a MSUT would seek measurement conditions andalgorithm settings to minimize the precision and offset of themeasurement. Minimization of TMU as described above, however, provides amore objective and comprehensive determination.

Turning to FIG. 7A, a flow diagram of a method of optimizing an MSUTaccording to a first embodiment is shown. In first step, S1, a structure8 (FIG. 3), i.e., artifact, is provided as described above relative tothe assessment method.

Step S2 (FIG. 7A) includes measuring a dimension of the plurality ofstructures according to a measurement parameter using a referencemeasurement system (RMS) to generate an RMS data set. A “measurementparameter” as used herein, refers to any measurement condition oranalysis parameter that affects the outcome of the measurement that canbe controllably altered. A “measurement parameter” may also include acombination of conditions and parameters or a variation of one of these.Measurement parameters may vary, for example, according to the type ofMSUT. For example, for an SEM, a measurement parameter may include atleast one of: a data smoothing amount, an algorithm setting, a beamlanding energy, a current, an edge detection algorithm, a scan rate,etc. For a scatterometer, a measurement parameter may include at leastone of: a spectra averaging timeframe, a spectra wavelength range, anangle of incidence, area of measurement, a density of selectedwavelengths, number of adjustable characteristics in a theoreticalmodel, etc. For an AFM, a measurement parameter may include at least oneof: a number of scans, a timeframe between scans, a scanning speed, adata smoothing amount, area of measurement, a tip shape, etc. A step ofselecting a measurement parameter(s) (not shown) may also be included inthe optimization method. Subsequently, an RMS uncertainty (U_(RMS)) iscalculated. This calculation may include calculation of an RMS precision(σ_(RMS)) according to any now known or later developed methodology,e.g., a standard deviation analysis. Alternatively, this calculation mayinclude calculating a TMU_(RMS) according to the methods disclosedherein. That is, the RMS may be treated as an MSUT and compared toanother RMS.

In next step, S3, measurement of the same dimension of the plurality ofstructures according to the same measurement parameter using the MSUT ismade to generate an MSUT data set. Subsequently, a precision of the MSUTfrom the MSUT data set is calculated.

Step S4 includes, as described above relative to the assessment method,conducting a Mandel linear regression analysis of the MSUT and RMS datasets to determine a corrected precision of the MSUT, and a net residualerror for the MSUT.

Next, at step S5, TMU is determined according to the formula:TMU=√{square root over (D _(M) ² −U _(RMS) ²)}  (19)where D_(M) is the Mandel net residual error (Eq. 12) and U_(RMS) is theRMS uncertainty, i.e., the RMS precision (σ_(RMS)) or an independentlydetermined TMU_(RMS). In other words, a TMU for the MSUT is determinedby removing the RMS uncertainty (U_(RMS)) from the net residual errorD_(M).

At step S6, a determination is made as to whether the determined TMU issubstantially different from the uncertainty for the MSUT (U_(MSUT)). Asnoted above, in a first cycle of steps S4-S5, the MSUT uncertainty isthe corrected precision. In subsequent cycles, the MSUT uncertainty is apreviously determined TMU value of the MSUT. If step S6 results in aYES, as discussed above, the Mandel linear regression may be repeatedwith the previously determined TMU value substituted for the MSUTuncertainty (U_(MSUT)) (step S7) in the ratio variable λ (Eq. 5). TheMandel linear regression analysis is preferably repeated until asufficient convergence of the MSUT uncertainty (U_(MSUT)) and TMU isachieved to declare a self-consistent result. What amounts to“sufficient convergence” or “substantially different” can be userdefined, e.g., as a percentage.

If the determination at step S6 is NO, then the determined TMU isconsidered the final TMU for that measurement parameter, and processingproceeds to step S8.

At step S8, a determination is made as to whether another measurementparameter (e.g., CD SEM smoothing filter adjustment) exists. If step S8results in YES, steps S3 to S7 may be repeated for another measurementparameter. The repeating step may recur for any number of measurementparameters. The resulting data includes a number of TMUs withcorresponding measurement parameter(s) and/or artifact structure(s). Ifstep S8 results in NO, processing proceeds to step S9.

Step S9 includes optimizing the MSUT by determining an optimalmeasurement parameter based on a minimal TMU. In particular, a minimalTMU is selected from a plurality of total measurement uncertainties of acorresponding plurality of measurement parameters. The correspondingmeasurement parameter represents the least imprecise and inaccurateenvironment for using the MSUT.

Referring to FIG. 7B, a flow diagram of a method for optimizing a MSUTaccording to a second embodiment is shown. This embodiment issubstantially similar to the embodiment of FIG. 7A, except that thelinear regression can be any general linear regression, e.g., an OLS. Inthis case, the TMU is defined according to the formula:TMU=√{square root over (D ² −U _(RMS) ²)}  (20)where D is the net residual error (Eq. 11) and U_(RMS) is the RMSuncertainty, i.e., the RMS precision (σ_(RMS)) or an independentlydetermined TMU_(RMS). In addition, the determined TMU for a measurementparameter in step S5 is considered the final TMU for that particularmeasurement parameter. Step S6 and S7 are identical to steps S8 and S9relative to the description of FIG. 7A.

Optimization Example

Referring to FIG. 8, an example derived from optimizing measurementconditions on a CDSEM for a resist isolated line geometry is graphicallyillustrated. The CD SEM starting conditions were those of one of the CDSEMs discussed earlier. While several acquisition conditions andalgorithms settings were optimized in this investigation, the graphshown in FIG. 8 illustrates the consequences of changing the amount ofsmoothing done to the raw CD SEM waveform prior to further algorithmanalysis. In particular, the noise reduction from this smoothing has apositive effect upon reducing the corrected precision. However, from thepoint of view of TMU, the trend is opposite. This suggests that the lossof accuracy in tracking the process changes in the artifact is worsewith greater smoothing as evidenced by this trend dominating the TMU.

IV. Conclusion

Although particular embodiments of assessment and optimization methodshave been described above, it should be recognized that particular stepsmay be omitted or varied. Accordingly, the invention should not belimited to any particular embodiment other than as provided in theattached claims.

In the previous discussion, it will be understood that the method stepsdiscussed may be performed by a processor executing instructions ofprogram product stored in a memory. It is understood that the variousdevices, modules, mechanisms and systems described herein may berealized in hardware, software, or a combination of hardware andsoftware, and may be compartmentalized other than as shown. They may beimplemented by any type of computer system or other apparatus adaptedfor carrying out the methods described herein. A typical combination ofhardware and software could be a general-purpose computer system with acomputer program that, when loaded and executed, controls the computersystem such that it carries out the methods described herein.Alternatively, a specific use computer, containing specialized hardwarefor carrying out one or more of the functional tasks of the inventioncould be utilized. The present invention can also be embedded in acomputer program product, which comprises all the features enabling theimplementation of the methods and functions described herein, andwhich—when loaded in a computer system—is able to carry out thesemethods and functions. Computer program, software program, program,program product, or software, in the present context mean anyexpression, in any language, code or notation, of a set of instructionsintended to cause a system having an information processing capabilityto perform a particular function either directly or after the following:(a) conversion to another language, code or notation; and/or (b)reproduction in a different material form.

While this invention has been described in conjunction with the specificembodiments outlined above, it is evident that many alternatives,modifications and variations will be apparent to those skilled in theart. Accordingly, the embodiments of the invention as set forth aboveare intended to be illustrative, not limiting. Various changes may bemade without departing from the spirit and scope of the invention asdefined in the following claims.

Industrial Applicability

The invention is useful for assessing and optimizing a measurementsystem under test.

1. A computer program product comprising a computer useable mediumhaving computer readable program code embodied therein for assessing ameasurement system under test (MSUT), the program product comprising:(a) program code configured to measure a dimension of the plurality ofstructures using a reference measurement system (RMS) to generate afirst data set, and calculate an RMS uncertainty (U_(RMS)) from thefirst data set, where the RMS uncertainty (U_(RMS)) is defined as one ofan RMS precision and an independently determined RMS total measurementuncertainty (TMU_(RMS)); (b) program code configured to measure thedimension of the plurality of structures using the MSUT to generate asecond data set, and calculate a precision of the MSUT from the seconddata set; (c) program code configured to conduct a linear regressionanalysis of the first and second data sets to determine a correctedprecision of the MSUT and a net residual error; (d) program codeconfigured to determine a total measurement uncertainty (TMU) for theMSUT by removing the RMS uncertainty (U_(RMS)) from the net residualerror; and (e) program code configured to output the TMU to a systemcapable of optimizing the MSUT.
 2. The program product of claim 1,wherein the code configured to determine the TMU implements the formula:TMU=√{square root over (D ² −U _(RMS) ²)} where D is the net residualerror.
 3. The program product of claim 1, wherein the code configured toconduct the linear regression implements a Mandel linear regressionwherein a ratio variable λ is defined according to the formula:$\lambda = \frac{U_{RMS}^{2}}{U_{MSUT}^{2}}$ where U_(MSUT) is as anMSUT uncertainty defined as one of the corrected precision of the MSUTand the TMU for the MSUT.
 4. The program product of claim 3, furthercomprising program code configured to, in the case that the TMU for theMSUT is substantially different than the MSUT uncertainty (U_(MSUT)),re-running the program code configured to conduct a linear regressionanalysis and the program code configured to determine TMU using the TMUfor the MSUT as the MSUT uncertainty (U_(MSUT)) in determining the ratiovariable λ.
 5. The program product of claim 3, wherein the codeconfigured to determine the TMU implements the formula:TMU=√{square root over (D _(M) ² −U _(RMS) ²)} where D_(M) is the Mandelnet residual error.
 6. A computer program product comprising a computeruseable medium having computer readable program code embodied thereinfor optimizing a measurement system under test (MSUT), the programproduct comprising: (a) program code configured to measure a dimensionof a plurality of structures according to a measurement parameter usinga reference measurement system (RMS) to generate a first data set, andcalculate an RMS uncertainty (U_(RMS)) from the first data set, wherethe RMS uncertainty (U_(RMS)) is defined as one of an RMS precision andan independently determined RMS total measurement uncertainty(TMU_(RMS)); (b) program code configured to measure the dimension of theplurality of structures according to the measurement parameter using theMSUT to generate a second data set, and calculate a precision of theMSUT from the second data set; (c) program code configured to conduct alinear regression analysis of the first and second data sets todetermine a corrected precision of the MSUT and a net residual error;(d) program code configured to determine a total measurement uncertainty(TMU) for the MSUT by removing the RMS uncertainty (URMS) from the netresidual error; (e) program code configured to output the TMU to asystem capable of optimizing the MSUT; and (f) program code configuredto optimize the MSUT by determining an optimal measurement parameterbased on a minimal total measurement uncertainty selected from aplurality of total measurement uncertainties of a correspondingplurality of measurement parameters.
 7. The program product of claim 6,wherein the code configured to determine the TMU implements the formula:TMU=√{square root over (D ² −U _(RMS) ²)} where D is the net residualerror.
 8. The program product of claim 6, wherein the code configured toconduct the linear regression implements a Mandel linear regressionwherein a ratio variable λ is defined according to the formula:$\lambda = \frac{U_{RMS}^{2}}{U_{MSUT}^{2}}$ where U_(MSUT) is as anMSUT uncertainty defined as one of the corrected precision of the MSUTand the TMU for the MSUT.
 9. The program product of claim 8, furthercomprising program code configured to, in the case that the TMU for theMSUT is substantially different than the MSUT uncertainty (U_(MSUT)),re-running the program code configured to conduct a linear regressionanalysis and the program code configured to determine TMU using the TMUfor the MSUT as the MSUT uncertainty (U_(MSUT)) in determining the ratiovariable λ.
 10. The program product of claim 8, wherein the codeconfigured to determine the TMU implements the formula:TMU=√{square root over (D _(M) ² −U _(RMS) ²)} where D_(M) is the Mandelnet residual error.